Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 72.7% → 81.2%

Time: 18.0s

Alternatives: 24

Speedup: 0.9×

• 57.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
  (* j (- (* c t) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
  (* j (- (* c t) (* i y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (+ (* x (- (* y z) (* t a))) (* b (- (* a i) (* z c))))
          (* j (- (* t c) (* y i))))))
   (if (<= t_1 INFINITY) t_1 (* j (fma c t (- 0.0 (* y i)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) + (b * ((a * i) - (z * c)))) + (j * ((t * c) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = j * fma(c, t, (0.0 - (y * i)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) + Float64(b * Float64(Float64(a * i) - Float64(z * c)))) + Float64(j * Float64(Float64(t * c) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(j * fma(c, t, Float64(0.0 - Float64(y * i))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 91.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 58.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 58.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right) \leq \infty:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + b \cdot \left(a \cdot i - z \cdot c\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, b \cdot i\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i 1.65e+66)
   (fma
    a
    (fma t (- 0.0 x) (* b i))
    (fma z (fma c (- 0.0 b) (* x y)) (* j (fma c t (- 0.0 (* y i))))))
   (fma i (fma j (- 0.0 y) (* a b)) (* x (- (* y z) (* t a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= 1.65e+66) {
		tmp = fma(a, fma(t, (0.0 - x), (b * i)), fma(z, fma(c, (0.0 - b), (x * y)), (j * fma(c, t, (0.0 - (y * i))))));
	} else {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (x * ((y * z) - (t * a))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= 1.65e+66)
		tmp = fma(a, fma(t, Float64(0.0 - x), Float64(b * i)), fma(z, fma(c, Float64(0.0 - b), Float64(x * y)), Float64(j * fma(c, t, Float64(0.0 - Float64(y * i))))));
	else
		tmp = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, 1.65e+66], N[(a * N[(t * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(z * N[(c * N[(0.0 - b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 1.6500000000000001e66

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 84.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   if 1.6500000000000001e66 < i

   1. Initial program 71.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, b \cdot i\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, x \cdot y\right), j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ t_2 := y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-212}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma i (* a b) (* x (- (* y z) (* t a)))))
        (t_2 (* y (fma j (- 0.0 i) (* x z)))))
   (if (<= y -3.1e+98)
     t_2
     (if (<= y -1.4e-112)
       t_1
       (if (<= y -1.25e-212)
         (* c (fma b (- 0.0 z) (* t j)))
         (if (<= y 1.6e-10) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(i, (a * b), (x * ((y * z) - (t * a))));
	double t_2 = y * fma(j, (0.0 - i), (x * z));
	double tmp;
	if (y <= -3.1e+98) {
		tmp = t_2;
	} else if (y <= -1.4e-112) {
		tmp = t_1;
	} else if (y <= -1.25e-212) {
		tmp = c * fma(b, (0.0 - z), (t * j));
	} else if (y <= 1.6e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(i, Float64(a * b), Float64(x * Float64(Float64(y * z) - Float64(t * a))))
	t_2 = Float64(y * fma(j, Float64(0.0 - i), Float64(x * z)))
	tmp = 0.0
	if (y <= -3.1e+98)
		tmp = t_2;
	elseif (y <= -1.4e-112)
		tmp = t_1;
	elseif (y <= -1.25e-212)
		tmp = Float64(c * fma(b, Float64(0.0 - z), Float64(t * j)));
	elseif (y <= 1.6e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e+98], t$95$2, If[LessEqual[y, -1.4e-112], t$95$1, If[LessEqual[y, -1.25e-212], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e-10], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.10000000000000019e98 or 1.5999999999999999e-10 < y

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

      9. *-lowering-*.f64 74.3

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(j, 0 - i, z \cdot x\right)} \]

   if -3.10000000000000019e98 < y < -1.40000000000000011e-112 or -1.25000000000000011e-212 < y < 1.5999999999999999e-10

   1. Initial program 82.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 70.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 65.3

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]

   8. Simplified 65.3%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]

   if -1.40000000000000011e-112 < y < -1.25000000000000011e-212

   1. Initial program 82.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      12. *-lowering-*.f64 74.5

          \[\leadsto c \cdot \mathsf{fma}\left(b, 0 - z, \color{blue}{j \cdot t}\right) \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, 0 - z, j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-212}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - t \cdot a\\ \mathbf{if}\;j \leq -2.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_1, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t \cdot c - y \cdot i, \frac{j}{z}, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* y z) (* t a))))
   (if (<= j -2.2e-69)
     (fma x t_1 (* j (fma c t (- 0.0 (* y i)))))
     (if (<= j 1.6e-49)
       (fma i (fma j (- 0.0 y) (* a b)) (* x t_1))
       (* z (fma (- (* t c) (* y i)) (/ j z) (* x y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (y * z) - (t * a);
	double tmp;
	if (j <= -2.2e-69) {
		tmp = fma(x, t_1, (j * fma(c, t, (0.0 - (y * i)))));
	} else if (j <= 1.6e-49) {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (x * t_1));
	} else {
		tmp = z * fma(((t * c) - (y * i)), (j / z), (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(y * z) - Float64(t * a))
	tmp = 0.0
	if (j <= -2.2e-69)
		tmp = fma(x, t_1, Float64(j * fma(c, t, Float64(0.0 - Float64(y * i)))));
	elseif (j <= 1.6e-49)
		tmp = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(x * t_1));
	else
		tmp = Float64(z * fma(Float64(Float64(t * c) - Float64(y * i)), Float64(j / z), Float64(x * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2.2e-69], N[(x * t$95$1 + N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.6e-49], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * N[(j / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.2e-69

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 74.8

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   if -2.2e-69 < j < 1.60000000000000001e-49

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   if 1.60000000000000001e-49 < j

   1. Initial program 74.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 73.7

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 73.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   8. Simplified 73.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y + \frac{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)}{z}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(x \cdot y + \frac{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)}{z}\right)} \]

       2. +-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(\frac{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)}{z} + x \cdot y\right)} \]

       3. *-commutative N/A

          \[\leadsto z \cdot \left(\frac{\color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right) \cdot j}}{z} + x \cdot y\right) \]

       4. associate-/l* N/A

          \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right) \cdot \frac{j}{z}} + x \cdot y\right) \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(-1 \cdot \left(i \cdot y\right) + c \cdot t, \frac{j}{z}, x \cdot y\right)} \]

       6. +-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{c \cdot t + -1 \cdot \left(i \cdot y\right)}, \frac{j}{z}, x \cdot y\right) \]

       7. mul-1-neg N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}, \frac{j}{z}, x \cdot y\right) \]

       8. unsub-neg N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{c \cdot t - i \cdot y}, \frac{j}{z}, x \cdot y\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{c \cdot t - i \cdot y}, \frac{j}{z}, x \cdot y\right) \]

       10. *-commutative N/A

           \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{t \cdot c} - i \cdot y, \frac{j}{z}, x \cdot y\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto z \cdot \mathsf{fma}\left(\color{blue}{t \cdot c} - i \cdot y, \frac{j}{z}, x \cdot y\right) \]

       12. *-commutative N/A

           \[\leadsto z \cdot \mathsf{fma}\left(t \cdot c - \color{blue}{y \cdot i}, \frac{j}{z}, x \cdot y\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto z \cdot \mathsf{fma}\left(t \cdot c - \color{blue}{y \cdot i}, \frac{j}{z}, x \cdot y\right) \]

       14. /-lowering-/.f64 N/A

           \[\leadsto z \cdot \mathsf{fma}\left(t \cdot c - y \cdot i, \color{blue}{\frac{j}{z}}, x \cdot y\right) \]

       15. *-lowering-*.f64 75.6

           \[\leadsto z \cdot \mathsf{fma}\left(t \cdot c - y \cdot i, \frac{j}{z}, \color{blue}{x \cdot y}\right) \]

   11. Simplified 75.6%

       \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(t \cdot c - y \cdot i, \frac{j}{z}, x \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.2 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.6 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t \cdot c - y \cdot i, \frac{j}{z}, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - t \cdot a\\ \mathbf{if}\;j \leq -6.3 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_1, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.25 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* y z) (* t a))))
   (if (<= j -6.3e-71)
     (fma x t_1 (* j (fma c t (- 0.0 (* y i)))))
     (if (<= j 1.25e-49)
       (fma i (fma j (- 0.0 y) (* a b)) (* x t_1))
       (- (* y (* x z)) (* j (- (* y i) (* t c))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (y * z) - (t * a);
	double tmp;
	if (j <= -6.3e-71) {
		tmp = fma(x, t_1, (j * fma(c, t, (0.0 - (y * i)))));
	} else if (j <= 1.25e-49) {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (x * t_1));
	} else {
		tmp = (y * (x * z)) - (j * ((y * i) - (t * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(y * z) - Float64(t * a))
	tmp = 0.0
	if (j <= -6.3e-71)
		tmp = fma(x, t_1, Float64(j * fma(c, t, Float64(0.0 - Float64(y * i)))));
	elseif (j <= 1.25e-49)
		tmp = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(x * t_1));
	else
		tmp = Float64(Float64(y * Float64(x * z)) - Float64(j * Float64(Float64(y * i) - Float64(t * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -6.3e-71], N[(x * t$95$1 + N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 1.25e-49], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision] - N[(j * N[(N[(y * i), $MachinePrecision] - N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -6.3000000000000003e-71

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 74.8

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   if -6.3000000000000003e-71 < j < 1.25e-49

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   if 1.25e-49 < j

   1. Initial program 74.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 75.4

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -6.3 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 1.25 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -4.3 \cdot 10^{+114}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -4.3e+114)
   (* j (- (* t c) (* y i)))
   (if (<= j 5.5e-50)
     (fma i (fma j (- 0.0 y) (* a b)) (* x (- (* y z) (* t a))))
     (- (* y (* x z)) (* j (- (* y i) (* t c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -4.3e+114) {
		tmp = j * ((t * c) - (y * i));
	} else if (j <= 5.5e-50) {
		tmp = fma(i, fma(j, (0.0 - y), (a * b)), (x * ((y * z) - (t * a))));
	} else {
		tmp = (y * (x * z)) - (j * ((y * i) - (t * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -4.3e+114)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	elseif (j <= 5.5e-50)
		tmp = fma(i, fma(j, Float64(0.0 - y), Float64(a * b)), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = Float64(Float64(y * Float64(x * z)) - Float64(j * Float64(Float64(y * i) - Float64(t * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -4.3e+114], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 5.5e-50], N[(i * N[(j * N[(0.0 - y), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision] - N[(j * N[(N[(y * i), $MachinePrecision] - N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -4.3000000000000001e114

   1. Initial program 73.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + -1 \cdot \left(i \cdot y\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot t} - i \cdot y\right) \]

      7. *-lowering-*.f64 81.9

         \[\leadsto j \cdot \left(c \cdot t - \color{blue}{i \cdot y}\right) \]

   8. Simplified 81.9%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

   if -4.3000000000000001e114 < j < 5.49999999999999975e-50

   1. Initial program 80.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   if 5.49999999999999975e-50 < j

   1. Initial program 74.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 75.4

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -4.3 \cdot 10^{+114}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{elif}\;j \leq 5.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, a \cdot b\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\\ t_2 := y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-211}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* a (fma t (- 0.0 x) (* b i))))
        (t_2 (* y (fma j (- 0.0 i) (* x z)))))
   (if (<= y -6.5e+61)
     t_2
     (if (<= y -4.9e-111)
       t_1
       (if (<= y -3.8e-211)
         (* c (fma b (- 0.0 z) (* t j)))
         (if (<= y 7.6e-10) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = a * fma(t, (0.0 - x), (b * i));
	double t_2 = y * fma(j, (0.0 - i), (x * z));
	double tmp;
	if (y <= -6.5e+61) {
		tmp = t_2;
	} else if (y <= -4.9e-111) {
		tmp = t_1;
	} else if (y <= -3.8e-211) {
		tmp = c * fma(b, (0.0 - z), (t * j));
	} else if (y <= 7.6e-10) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(a * fma(t, Float64(0.0 - x), Float64(b * i)))
	t_2 = Float64(y * fma(j, Float64(0.0 - i), Float64(x * z)))
	tmp = 0.0
	if (y <= -6.5e+61)
		tmp = t_2;
	elseif (y <= -4.9e-111)
		tmp = t_1;
	elseif (y <= -3.8e-211)
		tmp = Float64(c * fma(b, Float64(0.0 - z), Float64(t * j)));
	elseif (y <= 7.6e-10)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(a * N[(t * N[(0.0 - x), $MachinePrecision] + N[(b * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(j * N[(0.0 - i), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.5e+61], t$95$2, If[LessEqual[y, -4.9e-111], t$95$1, If[LessEqual[y, -3.8e-211], N[(c * N[(b * N[(0.0 - z), $MachinePrecision] + N[(t * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.6e-10], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.4999999999999996e61 or 7.5999999999999996e-10 < y

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto y \cdot \left(\color{blue}{\left(-1 \cdot i\right) \cdot j} + x \cdot z\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \left(\color{blue}{j \cdot \left(-1 \cdot i\right)} + x \cdot z\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot i, x \cdot z\right)} \]

      5. mul-1-neg N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(i\right)}, x \cdot z\right) \]

      6. neg-sub0 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, \color{blue}{0 - i}, x \cdot z\right) \]

      8. *-commutative N/A

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

      9. *-lowering-*.f64 72.9

         \[\leadsto y \cdot \mathsf{fma}\left(j, 0 - i, \color{blue}{z \cdot x}\right) \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(j, 0 - i, z \cdot x\right)} \]

   if -6.4999999999999996e61 < y < -4.90000000000000019e-111 or -3.80000000000000012e-211 < y < 7.5999999999999996e-10

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto a \cdot \color{blue}{\left(-1 \cdot \left(t \cdot x\right) + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto a \cdot \left(\color{blue}{\left(\mathsf{neg}\left(t \cdot x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto a \cdot \left(\color{blue}{t \cdot \left(\mathsf{neg}\left(x\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \color{blue}{\left(-1 \cdot x\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(b \cdot i\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(b \cdot i\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto a \cdot \left(t \cdot \left(-1 \cdot x\right) + \color{blue}{b \cdot i}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto a \cdot \color{blue}{\mathsf{fma}\left(t, -1 \cdot x, b \cdot i\right)} \]

      9. mul-1-neg N/A

         \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{\mathsf{neg}\left(x\right)}, b \cdot i\right) \]

      10. neg-sub0 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, \color{blue}{0 - x}, b \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

      13. *-lowering-*.f64 58.4

          \[\leadsto a \cdot \mathsf{fma}\left(t, 0 - x, \color{blue}{i \cdot b}\right) \]

   5. Simplified 58.4%

      \[\leadsto \color{blue}{a \cdot \mathsf{fma}\left(t, 0 - x, i \cdot b\right)} \]

   if -4.90000000000000019e-111 < y < -3.80000000000000012e-211

   1. Initial program 82.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t - b \cdot z\right)} \]

      2. sub-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t + \left(\mathsf{neg}\left(b \cdot z\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto c \cdot \left(j \cdot t + \color{blue}{-1 \cdot \left(b \cdot z\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right) + j \cdot t\right)} \]

      5. mul-1-neg N/A

         \[\leadsto c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} + j \cdot t\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto c \cdot \left(\color{blue}{b \cdot \left(\mathsf{neg}\left(z\right)\right)} + j \cdot t\right) \]

      7. mul-1-neg N/A

         \[\leadsto c \cdot \left(b \cdot \color{blue}{\left(-1 \cdot z\right)} + j \cdot t\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(b, -1 \cdot z, j \cdot t\right)} \]

      9. mul-1-neg N/A

         \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{neg}\left(z\right)}, j \cdot t\right) \]

      10. neg-sub0 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto c \cdot \mathsf{fma}\left(b, \color{blue}{0 - z}, j \cdot t\right) \]

      12. *-lowering-*.f64 74.5

          \[\leadsto c \cdot \mathsf{fma}\left(b, 0 - z, \color{blue}{j \cdot t}\right) \]

   5. Simplified 74.5%

      \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(b, 0 - z, j \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-111}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-211}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(b, 0 - z, t \cdot j\right)\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(t, 0 - x, b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(j, 0 - i, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -2.6 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 9 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j -2.6e-70)
   (fma x (* y z) (* j (fma c t (- 0.0 (* y i)))))
   (if (<= j 9e-63)
     (fma i (* a b) (* x (- (* y z) (* t a))))
     (- (* y (* x z)) (* j (- (* y i) (* t c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= -2.6e-70) {
		tmp = fma(x, (y * z), (j * fma(c, t, (0.0 - (y * i)))));
	} else if (j <= 9e-63) {
		tmp = fma(i, (a * b), (x * ((y * z) - (t * a))));
	} else {
		tmp = (y * (x * z)) - (j * ((y * i) - (t * c)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= -2.6e-70)
		tmp = fma(x, Float64(y * z), Float64(j * fma(c, t, Float64(0.0 - Float64(y * i)))));
	elseif (j <= 9e-63)
		tmp = fma(i, Float64(a * b), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = Float64(Float64(y * Float64(x * z)) - Float64(j * Float64(Float64(y * i) - Float64(t * c))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, -2.6e-70], N[(x * N[(y * z), $MachinePrecision] + N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 9e-63], N[(i * N[(a * b), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision] - N[(j * N[(N[(y * i), $MachinePrecision] - N[(t * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if j < -2.60000000000000002e-70

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 74.8

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 68.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   8. Simplified 68.0%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   if -2.60000000000000002e-70 < j < 8.9999999999999999e-63

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 67.2

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]

   8. Simplified 67.2%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]

   if 8.9999999999999999e-63 < j

   1. Initial program 74.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(x \cdot z\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      5. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

      6. *-lowering-*.f64 73.2

         \[\leadsto y \cdot \color{blue}{\left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]

   5. Simplified 73.2%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot x\right)} + j \cdot \left(c \cdot t - i \cdot y\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2.6 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 9 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) - j \cdot \left(y \cdot i - t \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, y \cdot z, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{if}\;j \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma x (* y z) (* j (fma c t (- 0.0 (* y i)))))))
   (if (<= j -1.75e-70)
     t_1
     (if (<= j 9.5e-50) (fma i (* a b) (* x (- (* y z) (* t a)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(x, (y * z), (j * fma(c, t, (0.0 - (y * i)))));
	double tmp;
	if (j <= -1.75e-70) {
		tmp = t_1;
	} else if (j <= 9.5e-50) {
		tmp = fma(i, (a * b), (x * ((y * z) - (t * a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(x, Float64(y * z), Float64(j * fma(c, t, Float64(0.0 - Float64(y * i)))))
	tmp = 0.0
	if (j <= -1.75e-70)
		tmp = t_1;
	elseif (j <= 9.5e-50)
		tmp = fma(i, Float64(a * b), Float64(x * Float64(Float64(y * z) - Float64(t * a))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision] + N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.75e-70], t$95$1, If[LessEqual[j, 9.5e-50], N[(i * N[(a * b), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -1.74999999999999987e-70 or 9.4999999999999993e-50 < j

   1. Initial program 77.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 74.3

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 70.7

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   8. Simplified 70.7%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z}, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right) \]

   if -1.74999999999999987e-70 < j < 9.4999999999999993e-50

   1. Initial program 79.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 65.6

         \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]

   8. Simplified 65.6%

      \[\leadsto \mathsf{fma}\left(i, \color{blue}{a \cdot b}, x \cdot \left(y \cdot z - t \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.75 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \mathbf{elif}\;j \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(i, a \cdot b, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z, j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;z \cdot \left(0 - b \cdot c\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* a b))))
   (if (<= y -4.8e+67)
     (* z (* x y))
     (if (<= y -2.5e-111)
       t_1
       (if (<= y -1.3e-211)
         (* z (- 0.0 (* b c)))
         (if (<= y 3.8e-29) t_1 (* j (- 0.0 (* y i)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -4.8e+67) {
		tmp = z * (x * y);
	} else if (y <= -2.5e-111) {
		tmp = t_1;
	} else if (y <= -1.3e-211) {
		tmp = z * (0.0 - (b * c));
	} else if (y <= 3.8e-29) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (a * b)
    if (y <= (-4.8d+67)) then
        tmp = z * (x * y)
    else if (y <= (-2.5d-111)) then
        tmp = t_1
    else if (y <= (-1.3d-211)) then
        tmp = z * (0.0d0 - (b * c))
    else if (y <= 3.8d-29) then
        tmp = t_1
    else
        tmp = j * (0.0d0 - (y * i))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -4.8e+67) {
		tmp = z * (x * y);
	} else if (y <= -2.5e-111) {
		tmp = t_1;
	} else if (y <= -1.3e-211) {
		tmp = z * (0.0 - (b * c));
	} else if (y <= 3.8e-29) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (a * b)
	tmp = 0
	if y <= -4.8e+67:
		tmp = z * (x * y)
	elif y <= -2.5e-111:
		tmp = t_1
	elif y <= -1.3e-211:
		tmp = z * (0.0 - (b * c))
	elif y <= 3.8e-29:
		tmp = t_1
	else:
		tmp = j * (0.0 - (y * i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(a * b))
	tmp = 0.0
	if (y <= -4.8e+67)
		tmp = Float64(z * Float64(x * y));
	elseif (y <= -2.5e-111)
		tmp = t_1;
	elseif (y <= -1.3e-211)
		tmp = Float64(z * Float64(0.0 - Float64(b * c)));
	elseif (y <= 3.8e-29)
		tmp = t_1;
	else
		tmp = Float64(j * Float64(0.0 - Float64(y * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (a * b);
	tmp = 0.0;
	if (y <= -4.8e+67)
		tmp = z * (x * y);
	elseif (y <= -2.5e-111)
		tmp = t_1;
	elseif (y <= -1.3e-211)
		tmp = z * (0.0 - (b * c));
	elseif (y <= 3.8e-29)
		tmp = t_1;
	else
		tmp = j * (0.0 - (y * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+67], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.5e-111], t$95$1, If[LessEqual[y, -1.3e-211], N[(z * N[(0.0 - N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-29], t$95$1, N[(j * N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.80000000000000004e67

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. *-lowering-*.f64 46.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -4.80000000000000004e67 < y < -2.5000000000000001e-111 or -1.3e-211 < y < 3.79999999999999976e-29

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 43.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 43.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 39.4

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 39.4%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if -2.5000000000000001e-111 < y < -1.3e-211

   1. Initial program 82.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. neg-sub0 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

      12. *-lowering-*.f64 49.2

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot c\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - b \cdot c\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(0 - b \cdot c\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \left(0 - \color{blue}{c \cdot b}\right) \]

      5. *-lowering-*.f64 49.0

         \[\leadsto z \cdot \left(0 - \color{blue}{c \cdot b}\right) \]

   8. Simplified 49.0%

      \[\leadsto z \cdot \color{blue}{\left(0 - c \cdot b\right)} \]

   if 3.79999999999999976e-29 < y

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 46.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto i \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(j\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(-1 \cdot j\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

      8. --lowering--.f64 42.4

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

   8. Simplified 42.4%

      \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(0 - j\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(i \cdot y\right) \cdot \left(0 - j\right)} \]

      2. sub0-neg N/A

         \[\leadsto \left(i \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)} \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot y\right) \cdot j}\right) \]

      6. *-lowering-*.f64 47.1

         \[\leadsto -\color{blue}{\left(i \cdot y\right)} \cdot j \]

   10. Applied egg-rr 47.1%

       \[\leadsto \color{blue}{-\left(i \cdot y\right) \cdot j} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;z \cdot \left(0 - b \cdot c\right)\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+64}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-212}:\\ \;\;\;\;0 - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* a b))))
   (if (<= y -7.5e+64)
     (* z (* x y))
     (if (<= y -1.52e-111)
       t_1
       (if (<= y -4.2e-212)
         (- 0.0 (* c (* z b)))
         (if (<= y 5.4e-30) t_1 (* j (- 0.0 (* y i)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -7.5e+64) {
		tmp = z * (x * y);
	} else if (y <= -1.52e-111) {
		tmp = t_1;
	} else if (y <= -4.2e-212) {
		tmp = 0.0 - (c * (z * b));
	} else if (y <= 5.4e-30) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (a * b)
    if (y <= (-7.5d+64)) then
        tmp = z * (x * y)
    else if (y <= (-1.52d-111)) then
        tmp = t_1
    else if (y <= (-4.2d-212)) then
        tmp = 0.0d0 - (c * (z * b))
    else if (y <= 5.4d-30) then
        tmp = t_1
    else
        tmp = j * (0.0d0 - (y * i))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -7.5e+64) {
		tmp = z * (x * y);
	} else if (y <= -1.52e-111) {
		tmp = t_1;
	} else if (y <= -4.2e-212) {
		tmp = 0.0 - (c * (z * b));
	} else if (y <= 5.4e-30) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (a * b)
	tmp = 0
	if y <= -7.5e+64:
		tmp = z * (x * y)
	elif y <= -1.52e-111:
		tmp = t_1
	elif y <= -4.2e-212:
		tmp = 0.0 - (c * (z * b))
	elif y <= 5.4e-30:
		tmp = t_1
	else:
		tmp = j * (0.0 - (y * i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(a * b))
	tmp = 0.0
	if (y <= -7.5e+64)
		tmp = Float64(z * Float64(x * y));
	elseif (y <= -1.52e-111)
		tmp = t_1;
	elseif (y <= -4.2e-212)
		tmp = Float64(0.0 - Float64(c * Float64(z * b)));
	elseif (y <= 5.4e-30)
		tmp = t_1;
	else
		tmp = Float64(j * Float64(0.0 - Float64(y * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (a * b);
	tmp = 0.0;
	if (y <= -7.5e+64)
		tmp = z * (x * y);
	elseif (y <= -1.52e-111)
		tmp = t_1;
	elseif (y <= -4.2e-212)
		tmp = 0.0 - (c * (z * b));
	elseif (y <= 5.4e-30)
		tmp = t_1;
	else
		tmp = j * (0.0 - (y * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+64], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.52e-111], t$95$1, If[LessEqual[y, -4.2e-212], N[(0.0 - N[(c * N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.4e-30], t$95$1, N[(j * N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.5000000000000005e64

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. *-lowering-*.f64 46.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -7.5000000000000005e64 < y < -1.51999999999999998e-111 or -4.1999999999999999e-212 < y < 5.39999999999999975e-30

   1. Initial program 83.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 43.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 43.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 39.4

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 39.4%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if -1.51999999999999998e-111 < y < -4.1999999999999999e-212

   1. Initial program 82.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(b \cdot c\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto z \cdot \left(x \cdot y + \color{blue}{-1 \cdot \left(b \cdot c\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \left(b \cdot c\right) + x \cdot y\right)} \]

      5. associate-*r* N/A

         \[\leadsto z \cdot \left(\color{blue}{\left(-1 \cdot b\right) \cdot c} + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \left(\color{blue}{c \cdot \left(-1 \cdot b\right)} + x \cdot y\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(c, -1 \cdot b, x \cdot y\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{\mathsf{neg}\left(b\right)}, x \cdot y\right) \]

      9. neg-sub0 N/A

         \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, \color{blue}{0 - b}, x \cdot y\right) \]

      11. *-commutative N/A

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

      12. *-lowering-*.f64 49.2

          \[\leadsto z \cdot \mathsf{fma}\left(c, 0 - b, \color{blue}{y \cdot x}\right) \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(c, 0 - b, y \cdot x\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(c \cdot z\right)\right) \cdot -1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(c \cdot z\right) \cdot b\right)} \cdot -1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot \left(z \cdot b\right)\right)} \cdot -1 \]

      4. *-commutative N/A

         \[\leadsto \left(c \cdot \color{blue}{\left(b \cdot z\right)}\right) \cdot -1 \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{c \cdot \left(\left(b \cdot z\right) \cdot -1\right)} \]

      6. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(-1 \cdot \left(b \cdot z\right)\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \left(b \cdot z\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto c \cdot \color{blue}{\left(\mathsf{neg}\left(b \cdot z\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto c \cdot \left(\mathsf{neg}\left(\color{blue}{z \cdot b}\right)\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \]

      11. mul-1-neg N/A

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(-1 \cdot b\right)}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto c \cdot \color{blue}{\left(z \cdot \left(-1 \cdot b\right)\right)} \]

      13. mul-1-neg N/A

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}\right) \]

      14. neg-sub0 N/A

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(0 - b\right)}\right) \]

      15. --lowering--.f64 45.0

          \[\leadsto c \cdot \left(z \cdot \color{blue}{\left(0 - b\right)}\right) \]

   8. Simplified 45.0%

      \[\leadsto \color{blue}{c \cdot \left(z \cdot \left(0 - b\right)\right)} \]

   if 5.39999999999999975e-30 < y

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 46.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto i \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(j\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(-1 \cdot j\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

      8. --lowering--.f64 42.4

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

   8. Simplified 42.4%

      \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(0 - j\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(i \cdot y\right) \cdot \left(0 - j\right)} \]

      2. sub0-neg N/A

         \[\leadsto \left(i \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)} \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot y\right) \cdot j}\right) \]

      6. *-lowering-*.f64 47.1

         \[\leadsto -\color{blue}{\left(i \cdot y\right)} \cdot j \]

   10. Applied egg-rr 47.1%

       \[\leadsto \color{blue}{-\left(i \cdot y\right) \cdot j} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+64}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -1.52 \cdot 10^{-111}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-212}:\\ \;\;\;\;0 - c \cdot \left(z \cdot b\right)\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-211}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* a b))))
   (if (<= y -3.5e+65)
     (* z (* x y))
     (if (<= y -2.9e-69)
       t_1
       (if (<= y -2.8e-211)
         (* j (* t c))
         (if (<= y 1.75e-29) t_1 (* j (- 0.0 (* y i)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -3.5e+65) {
		tmp = z * (x * y);
	} else if (y <= -2.9e-69) {
		tmp = t_1;
	} else if (y <= -2.8e-211) {
		tmp = j * (t * c);
	} else if (y <= 1.75e-29) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (a * b)
    if (y <= (-3.5d+65)) then
        tmp = z * (x * y)
    else if (y <= (-2.9d-69)) then
        tmp = t_1
    else if (y <= (-2.8d-211)) then
        tmp = j * (t * c)
    else if (y <= 1.75d-29) then
        tmp = t_1
    else
        tmp = j * (0.0d0 - (y * i))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -3.5e+65) {
		tmp = z * (x * y);
	} else if (y <= -2.9e-69) {
		tmp = t_1;
	} else if (y <= -2.8e-211) {
		tmp = j * (t * c);
	} else if (y <= 1.75e-29) {
		tmp = t_1;
	} else {
		tmp = j * (0.0 - (y * i));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (a * b)
	tmp = 0
	if y <= -3.5e+65:
		tmp = z * (x * y)
	elif y <= -2.9e-69:
		tmp = t_1
	elif y <= -2.8e-211:
		tmp = j * (t * c)
	elif y <= 1.75e-29:
		tmp = t_1
	else:
		tmp = j * (0.0 - (y * i))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(a * b))
	tmp = 0.0
	if (y <= -3.5e+65)
		tmp = Float64(z * Float64(x * y));
	elseif (y <= -2.9e-69)
		tmp = t_1;
	elseif (y <= -2.8e-211)
		tmp = Float64(j * Float64(t * c));
	elseif (y <= 1.75e-29)
		tmp = t_1;
	else
		tmp = Float64(j * Float64(0.0 - Float64(y * i)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (a * b);
	tmp = 0.0;
	if (y <= -3.5e+65)
		tmp = z * (x * y);
	elseif (y <= -2.9e-69)
		tmp = t_1;
	elseif (y <= -2.8e-211)
		tmp = j * (t * c);
	elseif (y <= 1.75e-29)
		tmp = t_1;
	else
		tmp = j * (0.0 - (y * i));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+65], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.9e-69], t$95$1, If[LessEqual[y, -2.8e-211], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-29], t$95$1, N[(j * N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.5000000000000001e65

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. *-lowering-*.f64 46.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -3.5000000000000001e65 < y < -2.8999999999999998e-69 or -2.7999999999999998e-211 < y < 1.7499999999999999e-29

   1. Initial program 82.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 46.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 41.5

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 41.5%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if -2.8999999999999998e-69 < y < -2.7999999999999998e-211

   1. Initial program 86.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 53.0

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. *-lowering-*.f64 30.8

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 30.8%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot t\right) \cdot j} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t\right) \cdot j} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot c\right)} \cdot j \]

      5. *-lowering-*.f64 37.0

         \[\leadsto \color{blue}{\left(t \cdot c\right)} \cdot j \]

   10. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\left(t \cdot c\right) \cdot j} \]

   if 1.7499999999999999e-29 < y

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 46.9

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto i \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(j\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(-1 \cdot j\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

      8. --lowering--.f64 42.4

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

   8. Simplified 42.4%

      \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(0 - j\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(i \cdot y\right) \cdot \left(0 - j\right)} \]

      2. sub0-neg N/A

         \[\leadsto \left(i \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)} \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot y\right) \cdot j}\right) \]

      6. *-lowering-*.f64 47.1

         \[\leadsto -\color{blue}{\left(i \cdot y\right)} \cdot j \]

   10. Applied egg-rr 47.1%

       \[\leadsto \color{blue}{-\left(i \cdot y\right) \cdot j} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+65}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-69}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-211}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-29}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0 - a, t, y \cdot z\right)\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-269}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2.7e+114)
   (* x (fma (- 0.0 a) t (* y z)))
   (if (<= x -7.6e-269)
     (* b (- (* a i) (* z c)))
     (if (<= x 1.6e+104)
       (* j (fma c t (- 0.0 (* y i))))
       (* x (- (* y z) (* t a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -2.7e+114) {
		tmp = x * fma((0.0 - a), t, (y * z));
	} else if (x <= -7.6e-269) {
		tmp = b * ((a * i) - (z * c));
	} else if (x <= 1.6e+104) {
		tmp = j * fma(c, t, (0.0 - (y * i)));
	} else {
		tmp = x * ((y * z) - (t * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2.7e+114)
		tmp = Float64(x * fma(Float64(0.0 - a), t, Float64(y * z)));
	elseif (x <= -7.6e-269)
		tmp = Float64(b * Float64(Float64(a * i) - Float64(z * c)));
	elseif (x <= 1.6e+104)
		tmp = Float64(j * fma(c, t, Float64(0.0 - Float64(y * i))));
	else
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2.7e+114], N[(x * N[(N[(0.0 - a), $MachinePrecision] * t + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.6e-269], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e+104], N[(j * N[(c * t + N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.7e114

   1. Initial program 89.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 89.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(t \cdot a\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(t \cdot a\right)\right) + y \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{a \cdot t}\right)\right) + y \cdot z\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot t} + y \cdot z\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, y \cdot z\right)} \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \]

      8. *-lowering-*.f64 89.1

         \[\leadsto x \cdot \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \]

   7. Applied egg-rr 89.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right)} \]

   if -2.7e114 < x < -7.6000000000000005e-269

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 56.5

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]

   if -7.6000000000000005e-269 < x < 1.6e104

   1. Initial program 68.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. sub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right) \]

      7. mul-1-neg N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      8. neg-sub0 N/A

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

      9. --lowering--.f64 59.3

         \[\leadsto j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right) \]

   5. Simplified 59.3%

      \[\leadsto \color{blue}{j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)} \]

   if 1.6e104 < x

   1. Initial program 79.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 77.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0 - a, t, y \cdot z\right)\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{-269}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \mathsf{fma}\left(c, t, 0 - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \mathsf{fma}\left(x, 0 - a, c \cdot j\right)\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-273}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+191}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* t (fma x (- 0.0 a) (* c j)))))
   (if (<= x -3.9e+126)
     t_1
     (if (<= x -5.8e-273)
       (* b (- (* a i) (* z c)))
       (if (<= x 6.5e+191) (* j (- (* t c) (* y i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = t * fma(x, (0.0 - a), (c * j));
	double tmp;
	if (x <= -3.9e+126) {
		tmp = t_1;
	} else if (x <= -5.8e-273) {
		tmp = b * ((a * i) - (z * c));
	} else if (x <= 6.5e+191) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(t * fma(x, Float64(0.0 - a), Float64(c * j)))
	tmp = 0.0
	if (x <= -3.9e+126)
		tmp = t_1;
	elseif (x <= -5.8e-273)
		tmp = Float64(b * Float64(Float64(a * i) - Float64(z * c)));
	elseif (x <= 6.5e+191)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(t * N[(x * N[(0.0 - a), $MachinePrecision] + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.9e+126], t$95$1, If[LessEqual[x, -5.8e-273], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e+191], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999993e126 or 6.50000000000000008e191 < x

   1. Initial program 81.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]

      2. mul-1-neg N/A

         \[\leadsto t \cdot \left(\color{blue}{\left(\mathsf{neg}\left(a \cdot x\right)\right)} + c \cdot j\right) \]

      3. *-commutative N/A

         \[\leadsto t \cdot \left(\left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) + c \cdot j\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto t \cdot \left(\color{blue}{x \cdot \left(\mathsf{neg}\left(a\right)\right)} + c \cdot j\right) \]

      5. mul-1-neg N/A

         \[\leadsto t \cdot \left(x \cdot \color{blue}{\left(-1 \cdot a\right)} + c \cdot j\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto t \cdot \color{blue}{\mathsf{fma}\left(x, -1 \cdot a, c \cdot j\right)} \]

      7. mul-1-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(a\right)}, c \cdot j\right) \]

      8. neg-sub0 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{0 - a}, c \cdot j\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{0 - a}, c \cdot j\right) \]

      10. *-commutative N/A

          \[\leadsto t \cdot \mathsf{fma}\left(x, 0 - a, \color{blue}{j \cdot c}\right) \]

      11. *-lowering-*.f64 67.6

          \[\leadsto t \cdot \mathsf{fma}\left(x, 0 - a, \color{blue}{j \cdot c}\right) \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{t \cdot \mathsf{fma}\left(x, 0 - a, j \cdot c\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{neg}\left(a\right)}, j \cdot c\right) \]

      2. neg-lowering-neg.f64 67.6

         \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{-a}, j \cdot c\right) \]

   10. Applied egg-rr 67.6%

       \[\leadsto t \cdot \mathsf{fma}\left(x, \color{blue}{-a}, j \cdot c\right) \]

   if -3.89999999999999993e126 < x < -5.79999999999999973e-273

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 56.5

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]

   if -5.79999999999999973e-273 < x < 6.50000000000000008e191

   1. Initial program 71.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + -1 \cdot \left(i \cdot y\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot t} - i \cdot y\right) \]

      7. *-lowering-*.f64 57.0

         \[\leadsto j \cdot \left(c \cdot t - \color{blue}{i \cdot y}\right) \]

   8. Simplified 57.0%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+126}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, 0 - a, c \cdot j\right)\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-273}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+191}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \mathsf{fma}\left(x, 0 - a, c \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0 - a, t, y \cdot z\right)\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-269}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x -2.5e+110)
   (* x (fma (- 0.0 a) t (* y z)))
   (if (<= x -9.5e-269)
     (* b (- (* a i) (* z c)))
     (if (<= x 1.4e+104)
       (* j (- (* t c) (* y i)))
       (* x (- (* y z) (* t a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= -2.5e+110) {
		tmp = x * fma((0.0 - a), t, (y * z));
	} else if (x <= -9.5e-269) {
		tmp = b * ((a * i) - (z * c));
	} else if (x <= 1.4e+104) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = x * ((y * z) - (t * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= -2.5e+110)
		tmp = Float64(x * fma(Float64(0.0 - a), t, Float64(y * z)));
	elseif (x <= -9.5e-269)
		tmp = Float64(b * Float64(Float64(a * i) - Float64(z * c)));
	elseif (x <= 1.4e+104)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	else
		tmp = Float64(x * Float64(Float64(y * z) - Float64(t * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, -2.5e+110], N[(x * N[(N[(0.0 - a), $MachinePrecision] * t + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-269], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e+104], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.49999999999999989e110

   1. Initial program 89.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 89.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z + \left(\mathsf{neg}\left(t \cdot a\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(t \cdot a\right)\right) + y \cdot z\right)} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{a \cdot t}\right)\right) + y \cdot z\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot t} + y \cdot z\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(a\right), t, y \cdot z\right)} \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(a\right)}, t, y \cdot z\right) \]

      7. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, \color{blue}{z \cdot y}\right) \]

      8. *-lowering-*.f64 89.1

         \[\leadsto x \cdot \mathsf{fma}\left(-a, t, \color{blue}{z \cdot y}\right) \]

   7. Applied egg-rr 89.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(-a, t, z \cdot y\right)} \]

   if -2.49999999999999989e110 < x < -9.5000000000000006e-269

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 56.5

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]

   if -9.5000000000000006e-269 < x < 1.4e104

   1. Initial program 68.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + -1 \cdot \left(i \cdot y\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot t} - i \cdot y\right) \]

      7. *-lowering-*.f64 58.2

         \[\leadsto j \cdot \left(c \cdot t - \color{blue}{i \cdot y}\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]

   if 1.4e104 < x

   1. Initial program 79.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 77.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0 - a, t, y \cdot z\right)\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-269}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-274}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (- (* y z) (* t a)))))
   (if (<= x -2.5e+110)
     t_1
     (if (<= x -2.6e-274)
       (* b (- (* a i) (* z c)))
       (if (<= x 1.25e+104) (* j (- (* t c) (* y i))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * ((y * z) - (t * a));
	double tmp;
	if (x <= -2.5e+110) {
		tmp = t_1;
	} else if (x <= -2.6e-274) {
		tmp = b * ((a * i) - (z * c));
	} else if (x <= 1.25e+104) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * z) - (t * a))
    if (x <= (-2.5d+110)) then
        tmp = t_1
    else if (x <= (-2.6d-274)) then
        tmp = b * ((a * i) - (z * c))
    else if (x <= 1.25d+104) then
        tmp = j * ((t * c) - (y * i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * ((y * z) - (t * a));
	double tmp;
	if (x <= -2.5e+110) {
		tmp = t_1;
	} else if (x <= -2.6e-274) {
		tmp = b * ((a * i) - (z * c));
	} else if (x <= 1.25e+104) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * ((y * z) - (t * a))
	tmp = 0
	if x <= -2.5e+110:
		tmp = t_1
	elif x <= -2.6e-274:
		tmp = b * ((a * i) - (z * c))
	elif x <= 1.25e+104:
		tmp = j * ((t * c) - (y * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(Float64(y * z) - Float64(t * a)))
	tmp = 0.0
	if (x <= -2.5e+110)
		tmp = t_1;
	elseif (x <= -2.6e-274)
		tmp = Float64(b * Float64(Float64(a * i) - Float64(z * c)));
	elseif (x <= 1.25e+104)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * ((y * z) - (t * a));
	tmp = 0.0;
	if (x <= -2.5e+110)
		tmp = t_1;
	elseif (x <= -2.6e-274)
		tmp = b * ((a * i) - (z * c));
	elseif (x <= 1.25e+104)
		tmp = j * ((t * c) - (y * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e+110], t$95$1, If[LessEqual[x, -2.6e-274], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25e+104], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.49999999999999989e110 or 1.2499999999999999e104 < x

   1. Initial program 83.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 81.8

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 81.8%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   if -2.49999999999999989e110 < x < -2.6e-274

   1. Initial program 84.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 82.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 56.5

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]

   if -2.6e-274 < x < 1.2499999999999999e104

   1. Initial program 68.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + -1 \cdot \left(i \cdot y\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot t} - i \cdot y\right) \]

      7. *-lowering-*.f64 58.2

         \[\leadsto j \cdot \left(c \cdot t - \color{blue}{i \cdot y}\right) \]

   8. Simplified 58.2%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-274}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := i \cdot \left(a \cdot b\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* i (* a b))))
   (if (<= y -2.6e+62)
     (* z (* x y))
     (if (<= y -2.3e-67)
       t_1
       (if (<= y -3.8e-212)
         (* j (* t c))
         (if (<= y 1.7e-22) t_1 (* x (* y z))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -2.6e+62) {
		tmp = z * (x * y);
	} else if (y <= -2.3e-67) {
		tmp = t_1;
	} else if (y <= -3.8e-212) {
		tmp = j * (t * c);
	} else if (y <= 1.7e-22) {
		tmp = t_1;
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = i * (a * b)
    if (y <= (-2.6d+62)) then
        tmp = z * (x * y)
    else if (y <= (-2.3d-67)) then
        tmp = t_1
    else if (y <= (-3.8d-212)) then
        tmp = j * (t * c)
    else if (y <= 1.7d-22) then
        tmp = t_1
    else
        tmp = x * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = i * (a * b);
	double tmp;
	if (y <= -2.6e+62) {
		tmp = z * (x * y);
	} else if (y <= -2.3e-67) {
		tmp = t_1;
	} else if (y <= -3.8e-212) {
		tmp = j * (t * c);
	} else if (y <= 1.7e-22) {
		tmp = t_1;
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = i * (a * b)
	tmp = 0
	if y <= -2.6e+62:
		tmp = z * (x * y)
	elif y <= -2.3e-67:
		tmp = t_1
	elif y <= -3.8e-212:
		tmp = j * (t * c)
	elif y <= 1.7e-22:
		tmp = t_1
	else:
		tmp = x * (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(i * Float64(a * b))
	tmp = 0.0
	if (y <= -2.6e+62)
		tmp = Float64(z * Float64(x * y));
	elseif (y <= -2.3e-67)
		tmp = t_1;
	elseif (y <= -3.8e-212)
		tmp = Float64(j * Float64(t * c));
	elseif (y <= 1.7e-22)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = i * (a * b);
	tmp = 0.0;
	if (y <= -2.6e+62)
		tmp = z * (x * y);
	elseif (y <= -2.3e-67)
		tmp = t_1;
	elseif (y <= -3.8e-212)
		tmp = j * (t * c);
	elseif (y <= 1.7e-22)
		tmp = t_1;
	else
		tmp = x * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+62], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.3e-67], t$95$1, If[LessEqual[y, -3.8e-212], N[(j * N[(t * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e-22], t$95$1, N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.59999999999999984e62

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. *-lowering-*.f64 46.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -2.59999999999999984e62 < y < -2.3e-67 or -3.80000000000000022e-212 < y < 1.6999999999999999e-22

   1. Initial program 82.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 45.5

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 45.5%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 40.3

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 40.3%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if -2.3e-67 < y < -3.80000000000000022e-212

   1. Initial program 86.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 53.0

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. *-lowering-*.f64 30.8

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 30.8%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \color{blue}{\left(t \cdot j\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(c \cdot t\right) \cdot j} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(c \cdot t\right) \cdot j} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(t \cdot c\right)} \cdot j \]

      5. *-lowering-*.f64 37.0

         \[\leadsto \color{blue}{\left(t \cdot c\right)} \cdot j \]

   10. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\left(t \cdot c\right) \cdot j} \]

   if 1.6999999999999999e-22 < y

   1. Initial program 78.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 43.9

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 36.7

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 36.7%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+62}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-67}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-212}:\\ \;\;\;\;j \cdot \left(t \cdot c\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-22}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{if}\;b \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-94}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* a i) (* z c)))))
   (if (<= b -8.8e+15)
     t_1
     (if (<= b 1.26e-94) (* j (- (* t c) (* y i))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -8.8e+15) {
		tmp = t_1;
	} else if (b <= 1.26e-94) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = b * ((a * i) - (z * c))
    if (b <= (-8.8d+15)) then
        tmp = t_1
    else if (b <= 1.26d-94) then
        tmp = j * ((t * c) - (y * i))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = b * ((a * i) - (z * c));
	double tmp;
	if (b <= -8.8e+15) {
		tmp = t_1;
	} else if (b <= 1.26e-94) {
		tmp = j * ((t * c) - (y * i));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = b * ((a * i) - (z * c))
	tmp = 0
	if b <= -8.8e+15:
		tmp = t_1
	elif b <= 1.26e-94:
		tmp = j * ((t * c) - (y * i))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(a * i) - Float64(z * c)))
	tmp = 0.0
	if (b <= -8.8e+15)
		tmp = t_1;
	elseif (b <= 1.26e-94)
		tmp = Float64(j * Float64(Float64(t * c) - Float64(y * i)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = b * ((a * i) - (z * c));
	tmp = 0.0;
	if (b <= -8.8e+15)
		tmp = t_1;
	elseif (b <= 1.26e-94)
		tmp = j * ((t * c) - (y * i));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.8e+15], t$95$1, If[LessEqual[b, 1.26e-94], N[(j * N[(N[(t * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.8e15 or 1.2600000000000001e-94 < b

   1. Initial program 79.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 61.9

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 61.9%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]

   if -8.8e15 < b < 1.2600000000000001e-94

   1. Initial program 77.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{j \cdot \left(-1 \cdot \left(i \cdot y\right) + c \cdot t\right)} \]

      2. +-commutative N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t + -1 \cdot \left(i \cdot y\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto j \cdot \left(c \cdot t + \color{blue}{\left(\mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto j \cdot \color{blue}{\left(c \cdot t - i \cdot y\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto j \cdot \left(\color{blue}{c \cdot t} - i \cdot y\right) \]

      7. *-lowering-*.f64 52.4

         \[\leadsto j \cdot \left(c \cdot t - \color{blue}{i \cdot y}\right) \]

   8. Simplified 52.4%

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{elif}\;b \leq 1.26 \cdot 10^{-94}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(0 - y \cdot i\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- 0.0 (* y i)))))
   (if (<= y -5.5e+161)
     t_1
     (if (<= y 7.5e+108) (* b (- (* a i) (* z c))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * (0.0 - (y * i));
	double tmp;
	if (y <= -5.5e+161) {
		tmp = t_1;
	} else if (y <= 7.5e+108) {
		tmp = b * ((a * i) - (z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = j * (0.0d0 - (y * i))
    if (y <= (-5.5d+161)) then
        tmp = t_1
    else if (y <= 7.5d+108) then
        tmp = b * ((a * i) - (z * c))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * (0.0 - (y * i));
	double tmp;
	if (y <= -5.5e+161) {
		tmp = t_1;
	} else if (y <= 7.5e+108) {
		tmp = b * ((a * i) - (z * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * (0.0 - (y * i))
	tmp = 0
	if y <= -5.5e+161:
		tmp = t_1
	elif y <= 7.5e+108:
		tmp = b * ((a * i) - (z * c))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(0.0 - Float64(y * i)))
	tmp = 0.0
	if (y <= -5.5e+161)
		tmp = t_1;
	elseif (y <= 7.5e+108)
		tmp = Float64(b * Float64(Float64(a * i) - Float64(z * c)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = j * (0.0 - (y * i));
	tmp = 0.0;
	if (y <= -5.5e+161)
		tmp = t_1;
	elseif (y <= 7.5e+108)
		tmp = b * ((a * i) - (z * c));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(0.0 - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+161], t$95$1, If[LessEqual[y, 7.5e+108], N[(b * N[(N[(a * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000005e161 or 7.50000000000000039e108 < y

   1. Initial program 70.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 62.8

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 62.8%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around inf

      \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto i \cdot \left(\mathsf{neg}\left(\color{blue}{y \cdot j}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(\mathsf{neg}\left(j\right)\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(-1 \cdot j\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(-1 \cdot j\right)\right)} \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)}\right) \]

      7. neg-sub0 N/A

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

      8. --lowering--.f64 54.2

         \[\leadsto i \cdot \left(y \cdot \color{blue}{\left(0 - j\right)}\right) \]

   8. Simplified 54.2%

      \[\leadsto i \cdot \color{blue}{\left(y \cdot \left(0 - j\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(i \cdot y\right) \cdot \left(0 - j\right)} \]

      2. sub0-neg N/A

         \[\leadsto \left(i \cdot y\right) \cdot \color{blue}{\left(\mathsf{neg}\left(j\right)\right)} \]

      3. distribute-rgt-neg-out N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(i \cdot y\right) \cdot j\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(i \cdot y\right) \cdot j}\right) \]

      6. *-lowering-*.f64 59.1

         \[\leadsto -\color{blue}{\left(i \cdot y\right)} \cdot j \]

   10. Applied egg-rr 59.1%

       \[\leadsto \color{blue}{-\left(i \cdot y\right) \cdot j} \]

   if -5.5000000000000005e161 < y < 7.50000000000000039e108

   1. Initial program 81.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(j \cdot \left(c \cdot t - i \cdot y\right) + z \cdot \left(x \cdot y - b \cdot c\right)\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]

   5. Simplified 81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(t, 0 - x, i \cdot b\right), \mathsf{fma}\left(z, \mathsf{fma}\left(c, 0 - b, y \cdot x\right), j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \left(c \cdot z\right) + a \cdot i\right)} \]

      2. +-commutative N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i + -1 \cdot \left(c \cdot z\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto b \cdot \left(a \cdot i + \color{blue}{\left(\mathsf{neg}\left(c \cdot z\right)\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      5. --lowering--.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(a \cdot i - c \cdot z\right)} \]

      6. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto b \cdot \left(\color{blue}{i \cdot a} - c \cdot z\right) \]

      8. *-lowering-*.f64 44.8

         \[\leadsto b \cdot \left(i \cdot a - \color{blue}{c \cdot z}\right) \]

   8. Simplified 44.8%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot a - c \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+161}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;b \cdot \left(a \cdot i - z \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(0 - y \cdot i\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+69}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-25}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -2.7e+69)
   (* z (* x y))
   (if (<= y 9.6e-25) (* i (* a b)) (* x (* y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -2.7e+69) {
		tmp = z * (x * y);
	} else if (y <= 9.6e-25) {
		tmp = i * (a * b);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (y <= (-2.7d+69)) then
        tmp = z * (x * y)
    else if (y <= 9.6d-25) then
        tmp = i * (a * b)
    else
        tmp = x * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -2.7e+69) {
		tmp = z * (x * y);
	} else if (y <= 9.6e-25) {
		tmp = i * (a * b);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -2.7e+69:
		tmp = z * (x * y)
	elif y <= 9.6e-25:
		tmp = i * (a * b)
	else:
		tmp = x * (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -2.7e+69)
		tmp = Float64(z * Float64(x * y));
	elseif (y <= 9.6e-25)
		tmp = Float64(i * Float64(a * b));
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -2.7e+69)
		tmp = z * (x * y);
	elseif (y <= 9.6e-25)
		tmp = i * (a * b);
	else
		tmp = x * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -2.7e+69], N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e-25], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999998e69

   1. Initial program 65.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 51.5

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 51.5%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot x \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot y\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \left(x \cdot y\right)} \]

      6. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

      7. *-lowering-*.f64 46.0

         \[\leadsto z \cdot \color{blue}{\left(y \cdot x\right)} \]

   8. Simplified 46.0%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -2.6999999999999998e69 < y < 9.60000000000000037e-25

   1. Initial program 83.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 37.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 37.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 32.7

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 32.7%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   if 9.60000000000000037e-25 < y

   1. Initial program 78.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 43.9

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 36.7

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 36.7%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+69}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-25}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-21}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= y -8e+63) t_1 (if (<= y 1.1e-21) (* i (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -8e+63) {
		tmp = t_1;
	} else if (y <= 1.1e-21) {
		tmp = i * (a * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y * z)
    if (y <= (-8d+63)) then
        tmp = t_1
    else if (y <= 1.1d-21) then
        tmp = i * (a * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = x * (y * z);
	double tmp;
	if (y <= -8e+63) {
		tmp = t_1;
	} else if (y <= 1.1e-21) {
		tmp = i * (a * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if y <= -8e+63:
		tmp = t_1
	elif y <= 1.1e-21:
		tmp = i * (a * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(x * Float64(y * z))
	tmp = 0.0
	if (y <= -8e+63)
		tmp = t_1;
	elseif (y <= 1.1e-21)
		tmp = Float64(i * Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = x * (y * z);
	tmp = 0.0;
	if (y <= -8e+63)
		tmp = t_1;
	elseif (y <= 1.1e-21)
		tmp = i * (a * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+63], t$95$1, If[LessEqual[y, 1.1e-21], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000046e63 or 1.1e-21 < y

   1. Initial program 73.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]

      2. --lowering--.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z - a \cdot t\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot z} - a \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

      5. *-lowering-*.f64 47.0

         \[\leadsto x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) \]

   5. Simplified 47.0%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 40.5

         \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   8. Simplified 40.5%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]

   if -8.00000000000000046e63 < y < 1.1e-21

   1. Initial program 83.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 37.3

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 37.3%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 32.7

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 32.7%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-21}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -2 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 3.3 \cdot 10^{-118}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -2e+98) t_1 (if (<= j 3.3e-118) (* i (* a b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2e+98) {
		tmp = t_1;
	} else if (j <= 3.3e-118) {
		tmp = i * (a * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-2d+98)) then
        tmp = t_1
    else if (j <= 3.3d-118) then
        tmp = i * (a * b)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -2e+98) {
		tmp = t_1;
	} else if (j <= 3.3e-118) {
		tmp = i * (a * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -2e+98:
		tmp = t_1
	elif j <= 3.3e-118:
		tmp = i * (a * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -2e+98)
		tmp = t_1;
	elseif (j <= 3.3e-118)
		tmp = Float64(i * Float64(a * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -2e+98)
		tmp = t_1;
	elseif (j <= 3.3e-118)
		tmp = i * (a * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -2e+98], t$95$1, If[LessEqual[j, 3.3e-118], N[(i * N[(a * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -2e98 or 3.3e-118 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 72.7

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. *-lowering-*.f64 37.6

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

   if -2e98 < j < 3.3e-118

   1. Initial program 81.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto i \cdot \left(\color{blue}{\left(\mathsf{neg}\left(j \cdot y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto i \cdot \left(\color{blue}{j \cdot \left(\mathsf{neg}\left(y\right)\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \color{blue}{\left(-1 \cdot y\right)} + \left(\mathsf{neg}\left(-1 \cdot \left(a \cdot b\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)}\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto i \cdot \left(j \cdot \left(-1 \cdot y\right) + \color{blue}{a \cdot b}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto i \cdot \color{blue}{\mathsf{fma}\left(j, -1 \cdot y, a \cdot b\right)} \]

      9. mul-1-neg N/A

         \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{\mathsf{neg}\left(y\right)}, a \cdot b\right) \]

      10. neg-sub0 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, \color{blue}{0 - y}, a \cdot b\right) \]

      12. *-commutative N/A

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

      13. *-lowering-*.f64 40.1

          \[\leadsto i \cdot \mathsf{fma}\left(j, 0 - y, \color{blue}{b \cdot a}\right) \]

   5. Simplified 40.1%

      \[\leadsto \color{blue}{i \cdot \mathsf{fma}\left(j, 0 - y, b \cdot a\right)} \]

   6. Taylor expanded in j around 0

      \[\leadsto i \cdot \color{blue}{\left(a \cdot b\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

      2. *-lowering-*.f64 30.6

         \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]

   8. Simplified 30.6%

      \[\leadsto i \cdot \color{blue}{\left(b \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -2 \cdot 10^{+98}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 3.3 \cdot 10^{-118}:\\ \;\;\;\;i \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 28.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(t \cdot j\right)\\ \mathbf{if}\;j \leq -5.4 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 2.35 \cdot 10^{-118}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (* t j))))
   (if (<= j -5.4e+97) t_1 (if (<= j 2.35e-118) (* a (* b i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -5.4e+97) {
		tmp = t_1;
	} else if (j <= 2.35e-118) {
		tmp = a * (b * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = c * (t * j)
    if (j <= (-5.4d+97)) then
        tmp = t_1
    else if (j <= 2.35d-118) then
        tmp = a * (b * i)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = c * (t * j);
	double tmp;
	if (j <= -5.4e+97) {
		tmp = t_1;
	} else if (j <= 2.35e-118) {
		tmp = a * (b * i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * (t * j)
	tmp = 0
	if j <= -5.4e+97:
		tmp = t_1
	elif j <= 2.35e-118:
		tmp = a * (b * i)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(t * j))
	tmp = 0.0
	if (j <= -5.4e+97)
		tmp = t_1;
	elseif (j <= 2.35e-118)
		tmp = Float64(a * Float64(b * i));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * (t * j);
	tmp = 0.0;
	if (j <= -5.4e+97)
		tmp = t_1;
	elseif (j <= 2.35e-118)
		tmp = a * (b * i);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(c * N[(t * j), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -5.4e+97], t$95$1, If[LessEqual[j, 2.35e-118], N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -5.39999999999999987e97 or 2.34999999999999995e-118 < j

   1. Initial program 74.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + j \cdot \left(c \cdot t - i \cdot y\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z - a \cdot t}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot z} - a \cdot t, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - \color{blue}{t \cdot a}, j \cdot \left(c \cdot t - i \cdot y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, \color{blue}{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\left(c \cdot t + \left(\mathsf{neg}\left(i \cdot y\right)\right)\right)}\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \color{blue}{\mathsf{fma}\left(c, t, \mathsf{neg}\left(i \cdot y\right)\right)}\right) \]

      10. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(-1 \cdot y\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, \color{blue}{i \cdot \left(-1 \cdot y\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      14. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

      15. --lowering--.f64 72.7

          \[\leadsto \mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \color{blue}{\left(0 - y\right)}\right)\right) \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot z - t \cdot a, j \cdot \mathsf{fma}\left(c, t, i \cdot \left(0 - y\right)\right)\right)} \]

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

      2. *-lowering-*.f64 37.6

         \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]

   8. Simplified 37.6%

      \[\leadsto \color{blue}{c \cdot \left(j \cdot t\right)} \]

   if -5.39999999999999987e97 < j < 2.34999999999999995e-118

   1. Initial program 81.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

      3. cancel-sign-sub N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

      10. cancel-sign-sub N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

      11. mul-1-neg N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

      12. associate-*r* N/A

          \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

      13. +-commutative N/A

          \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

   5. Simplified 71.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

      2. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

      3. *-lowering-*.f64 27.3

         \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   8. Simplified 27.3%

      \[\leadsto \color{blue}{a \cdot \left(i \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -5.4 \cdot 10^{+97}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \mathbf{elif}\;j \leq 2.35 \cdot 10^{-118}:\\ \;\;\;\;a \cdot \left(b \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(t \cdot j\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 22.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = a * (b * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - -1 \cdot \left(a \cdot \left(b \cdot i\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(-1 \cdot a\right) \cdot \left(b \cdot i\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) - \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot \left(b \cdot i\right) \]

   3. cancel-sign-sub N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)\right) + a \cdot \left(b \cdot i\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - a \cdot t\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)} + a \cdot \left(b \cdot i\right) \]

   5. associate-+l+ N/A

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right) + a \cdot \left(b \cdot i\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(-1 \cdot \color{blue}{\left(\left(j \cdot y\right) \cdot i\right)} + a \cdot \left(b \cdot i\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\color{blue}{\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i} + a \cdot \left(b \cdot i\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \left(\left(-1 \cdot \left(j \cdot y\right)\right) \cdot i + \color{blue}{\left(a \cdot b\right) \cdot i}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + a \cdot b\right)} \]

   10. cancel-sign-sub N/A

       \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \color{blue}{\left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \]

   11. mul-1-neg N/A

       \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{\left(-1 \cdot a\right)} \cdot b\right) \]

   12. associate-*r* N/A

       \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) + i \cdot \left(-1 \cdot \left(j \cdot y\right) - \color{blue}{-1 \cdot \left(a \cdot b\right)}\right) \]

   13. +-commutative N/A

       \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(a \cdot b\right)\right) + x \cdot \left(y \cdot z - a \cdot t\right)} \]

5. Simplified 64.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, \mathsf{fma}\left(j, 0 - y, b \cdot a\right), x \cdot \left(y \cdot z - t \cdot a\right)\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot \left(b \cdot i\right)} \]

   2. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

   3. *-lowering-*.f64 21.6

      \[\leadsto a \cdot \color{blue}{\left(i \cdot b\right)} \]

8. Simplified 21.6%

   \[\leadsto \color{blue}{a \cdot \left(i \cdot b\right)} \]

9. Final simplification 21.6%

   \[\leadsto a \cdot \left(b \cdot i\right) \]
10. Add Preprocessing

Developer Target 1: 67.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i, j)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))